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Properties

Label 2-2e4-16.13-c9-0-16
Degree $2$
Conductor $16$
Sign $-0.590 - 0.806i$
Analytic cond. $8.24057$
Root an. cond. $2.87063$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.39 − 21.3i)2-s + (−180. − 180. i)3-s + (−402. − 316. i)4-s + (1.44e3 − 1.44e3i)5-s + (−5.20e3 + 2.53e3i)6-s + 567. i·7-s + (−9.74e3 + 6.27e3i)8-s + 4.57e4i·9-s + (−2.02e4 − 4.15e4i)10-s + (4.11e4 − 4.11e4i)11-s + (1.56e4 + 1.30e5i)12-s + (5.37e3 + 5.37e3i)13-s + (1.21e4 + 4.20e3i)14-s − 5.22e5·15-s + (6.20e4 + 2.54e5i)16-s + 2.04e5·17-s + ⋯
L(s)  = 1  + (0.326 − 0.945i)2-s + (−1.28 − 1.28i)3-s + (−0.786 − 0.617i)4-s + (1.03 − 1.03i)5-s + (−1.64 + 0.797i)6-s + 0.0894i·7-s + (−0.840 + 0.541i)8-s + 2.32i·9-s + (−0.639 − 1.31i)10-s + (0.846 − 0.846i)11-s + (0.217 + 1.81i)12-s + (0.0521 + 0.0521i)13-s + (0.0845 + 0.0292i)14-s − 2.66·15-s + (0.236 + 0.971i)16-s + 0.594·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.590 - 0.806i$
Analytic conductor: \(8.24057\)
Root analytic conductor: \(2.87063\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :9/2),\ -0.590 - 0.806i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.518356 + 1.02167i\)
\(L(\frac12)\) \(\approx\) \(0.518356 + 1.02167i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.39 + 21.3i)T \)
good3 \( 1 + (180. + 180. i)T + 1.96e4iT^{2} \)
5 \( 1 + (-1.44e3 + 1.44e3i)T - 1.95e6iT^{2} \)
7 \( 1 - 567. iT - 4.03e7T^{2} \)
11 \( 1 + (-4.11e4 + 4.11e4i)T - 2.35e9iT^{2} \)
13 \( 1 + (-5.37e3 - 5.37e3i)T + 1.06e10iT^{2} \)
17 \( 1 - 2.04e5T + 1.18e11T^{2} \)
19 \( 1 + (6.21e5 + 6.21e5i)T + 3.22e11iT^{2} \)
23 \( 1 - 2.30e5iT - 1.80e12T^{2} \)
29 \( 1 + (-2.15e6 - 2.15e6i)T + 1.45e13iT^{2} \)
31 \( 1 + 6.21e6T + 2.64e13T^{2} \)
37 \( 1 + (5.11e6 - 5.11e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 7.09e6iT - 3.27e14T^{2} \)
43 \( 1 + (-8.65e5 + 8.65e5i)T - 5.02e14iT^{2} \)
47 \( 1 - 3.03e6T + 1.11e15T^{2} \)
53 \( 1 + (-2.97e6 + 2.97e6i)T - 3.29e15iT^{2} \)
59 \( 1 + (4.44e6 - 4.44e6i)T - 8.66e15iT^{2} \)
61 \( 1 + (1.28e8 + 1.28e8i)T + 1.16e16iT^{2} \)
67 \( 1 + (-1.33e8 - 1.33e8i)T + 2.72e16iT^{2} \)
71 \( 1 + 2.06e8iT - 4.58e16T^{2} \)
73 \( 1 + 2.14e8iT - 5.88e16T^{2} \)
79 \( 1 - 6.09e8T + 1.19e17T^{2} \)
83 \( 1 + (1.95e8 + 1.95e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 6.63e8iT - 3.50e17T^{2} \)
97 \( 1 + 4.16e8T + 7.60e17T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.78231830806796670119050471844, −13.88053427347611624823160210793, −12.98043345580300101906607921528, −12.08836673408723669524600353250, −10.83158874554659954709123911567, −8.917414819576331618012491696803, −6.24641391670928735849797100190, −5.14856132644549897292403028608, −1.74739822574021168424245021069, −0.64793323683852294312269405488, 4.01724993114074360191655324937, 5.66673857931739754816356353967, 6.64529213819609468288037089223, 9.524951563018473181096508550665, 10.51005449039154950607614583882, 12.26546112456446244383517900701, 14.35389447787276147443108425835, 15.13887335481291532969053517077, 16.63530005213110652231099809909, 17.33292558354733630145303738943

Graph of the $Z$-function along the critical line